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Full text of "Analytical Mechanics"

144
ANALYTICAL MECHANICS
3. Find the center of mass of a semicircular lamina. Selecting the coordinates and the element of mass as shown in Fig.' we have
dm = cr-pd6 ğdp,
nay • crp dp d6 _         >_____________
V~    C'C'rpdpdB
Ğ/0    J 0
O
FIG. 76.
= 0.
PROBLEMS.
Find the center of mass of the lamina bounded by the following curves:
(1)  y = mx, y — — mx, and y — a.
(2)  y = asi&x, y = 0, x = 0, and x = TT.
(3)  yz = ax and #2 = by.
(4)  x2 + 7/2 = a2, Z = 0, and y = 0.
(5)  bV + a22/2 = a262, x = 0, and a/ = 0.
(6)  r=a(l+cos0).
(7)  r = a, 0 = 0, and 6 = 0.
(8)  r = a, r = b, 6 = 0, anc
7T
V
116. Center of Mass of a Homogeneous Solid of Revolution. — Let Fig. 77 represent any solid . obtained by revolving a plane curve about the x-axis. Then the center of mass lies on the axis of revolution. The position of the center of mass is found most conveniently when the element of mass is a thin slice obtained by two transverse sections.. The expression for the mass of such an element is
dm = r • wy2 - dx,
where r is the density of the solid, y the radius of the slice,, and dx its thickness.